st alg? assm + ncsm talks 2013

7/28/2019 St Alg? Assm + Ncsm Talks 2013

1/21

TeachingPhasesandStandardAlgorithms

intheCommonCoreStateStandards

orFromStrategiestoVariations

inWritingtheStandardAlgorithms

KarenC.Fuson

ProfessorEmerita,NorthwesternUniversity

SybillaBeckmann

Professor,UniversityofGeorgia

Thistalkisbasedon

theCCSS-Mstandards,

TheNBTProgressionfortheCommonCoreState

StandardsbyTheCommonCoreWritingTeam(7April,

2011),commoncoretools.wordpress.com,and

Fuson,K.C.&Beckmann,S.(Fall/Winter,2012-2013).

StandardalgorithmsintheCommonCoreState

Standards.NationalCouncilofSupervisorsofMathematicsJournalofMathematicsEducation

Leadership,14(2),14-30(whichispostedat

http://www.math.uga.edu/~sybilla/asisthistalkfile).


2/21

Learning Path Teaching-Learning:

Differentiating within Whole-Class Instruction by Using

the Math Talk Community

Bridgingforteachersandstudents Learning

bycoherentlearningsupports Path

Phase 3: Compact methods forfluency

Phase 2: Research-based mathematically-desirable and

accessible methods in the middle for

understanding and growing fluency

Phase 1: Students methods elicited forunderstandingbut move rapidly to Phase 2

Common Core Mathematical PracticesMath Sense-Making about Math Structure using Math Drawings to support Math Explaining

Math Sense-Making: Making sense and using appropriate precision

1 Make sense of problems and persevere in solving them.

6 Attend to precision.

Math Structure: Seeing structure and generalizing

7 Look for and make use of structure.

8 Look for and express regularity in repeated reasoning.

Math Drawings: Modeling and using tools

4 Model with mathematics.

5 Use appropriate tools strategically.

Math Explaining: Reasoning and explaining

2 Reason abstractly and quantitatively.

3 Construct viable arguments and critique the reasoning of others.

The top is an extension of Fuson, K. C. & Murata, A. (2007). Integrating NRC principles and the NCTM

Process Standards to form a Class Learning Path Model that individualizes within whole-class activities.

National Council of Supervisors of Mathematics Journal of Mathematics Education Leadership, 10 (1), 72-91.

It is a summary of several National Research Council Reports.

Math Drawings

Math Explaining

Math Drawings

Math Explaining

Math Sense-Making

Math Structure

Math Sense-Making

Math Structure


3/21

Themeaningfuldevelopmentofstandard

algorithmsintheCCSS-M

TheCCSS-Mconceptualapproachtocomputationis

deeplymathematicalandenablesstudentstomake

senseofandusethebasetensystemandpropertiesof

operationspowerfully.TheCCSS-Mfocuson

understandingandexplainingsuchcalculations,with

thesupportofvisualmodels,enablesstudentstoseemathematicalstructureasaccessible,important,

interesting,anduseful.

Therelationshipsacrossoperationsarealsoacritically

importantmathematicalidea.Howtheregularityofthe

mathematicalstructureinthebasetensystemcanbe

usedforsomanydifferentkindsofcalculationisan

importantfeatureofwhatwewantstudentsto

appreciateintheelementarygrades.

ItiscrucialtousetheStandardsofMathematical

Practicethroughoutthedevelopmentofcomputational

methods.


4/21

MisconceptionsabouttheCCSS-MandtheNBT

Progression.Theseareallwrong.

ThestandardalgorithmisthemethodIlearned.

Thestandardalgorithmisthemethodcommonlytaught

now(thecurrentcommonmethod).

Thereisonlyonewaytowritethealgorithmforeach

operation.

Thestandardalgorithmmeansteachingrotelywithout

understanding.

Teachersorprogramsmaynotteachthestandard

algorithmuntilthegradeatwhichfluencyisspecifiedin

theCCSS-M.

Initiallyteachersorprogramsmayonlyusemethods

thatchildreninvent.

Teachersorprogramsmustemphasizespecialstrategies

usefulonlyforcertainnumbers.


5/21

Generalmethodsthatwillgeneralizetoandbecome

standardalgorithmscanandshouldbedeveloped,

discussed,andexplainedinitiallyusingavisualmodel.

Thecriticalareafortheinitialgradeinwhichatypeof

multidigitcomputationisintroducedspecifiesthat:

Studentsdevelop,discuss,anduseefficient,

accurate,andgeneralizablemethodsto[+-x].

Thestandardfortheinitialgradeinwhichatypeofmultidigitcomputationisintroducedspecifiesthat

studentsareinitiallyto:

useconcretemodelsordrawingsandstrategiesbased

onplacevalue,propertiesofoperations,and/orthe

relationshipbetweenadditionandsubtraction;relate

thestrategytoawrittenmethodandexplainthe

reasoningused.[Grade1additionandGrade2

additionandsubtraction]

or

Illustrateandexplainthecalculationbyusing

equations,rectangulararrays,and/orareamodels.

[Grade4multiplicationanddivision]


6/21

Inthepast,therehasbeenanunfortunateandharmful

dichotomysuggestingthat

strategyimpliesunderstandingandalgorithmimpliesnovisualmodels,noexplaining,andnounderstanding.

Inthepast,teachingthestandardalgorithmhastoo

oftenmeantteachingnumericalstepsrotelyandhaving

studentsmemorizethestepsratherthanunderstand

andexplainthem.

TheCCSS-Mclearlydonotmeanforthistohappen,and

theNBTProgressiondocumentclarifiesthisbyshowing

visualmodelsandexplanationsforvariousminor

variationsinthewrittenmethodsforthestandard

algorithmsforalloperations.

Thewordstrategyemphasizesthatcomputationis

beingapproachedthoughtfullywithanemphasison

studentsense-making.Computationstrategyas

definedintheGlossaryfortheCCSS-Mincludesspecial

strategieschosenforspecificproblems,soastrategy

doesnothavetogeneralize.Buttheemphasisateverygradelevelwithinallofthecomputationstandardsis

onefficientandgeneralizablemethods,asintheCritical

Areas.


7/21

TheNBTProgressiondocumentsummarizesthat

thestandardalgorithmforanoperationimplements

thefollowingmathematicalapproachwithminorvariationsinhowthealgorithmiswritten:

decomposenumbersintobase-tenunitsandthen

carryingoutsingle-digitcomputationswiththose

unitsusingtheplacevaluestodirecttheplacevalue

oftheresultingnumber;and

usetheone-to-tenuniformityofthebaseten

structureofthenumbersystemtogeneralizetolarge

wholenumbersandtodecimals.

Toimplementastandardalgorithmoneusesa

systematicwrittenmethodforrecordingthestepsof

thealgorithm.

Therearevariationsinthesewrittenmethods

withinacountry

acrosscountries

atdifferenttimes.


8/21

CriteriaforEmphasizedWrittenMethods

ThatShouldbeIntroducedintheClassroom

Variationsthatsupportanduseplacevaluecorrectly

Variationsthatmakesingle-digitcomputationseasier,

giventhecentralityofsingle-digitcomputationsin

algorithms

Variationsinwhichallofonekindofstepisdonefirstandthentheotherkindofstepisdoneratherthan

alternating,becausevariationsinwhichthekindsof

stepsalternatecanintroduceerrorsandbemore

difficult.

Variationsthatkeeptheinitialmultidigitnumbers

unchangedbecausetheyareconceptuallyclearer

Variationsthatcanbedonelefttorightarehelpfulto

manystudentsbecausemanystudentspreferto

calculatefromlefttoright.


9/21

TheLearningPathUsingHelpingStepVariations

Somevariationsofawrittenmethodincludestepsor

mathdrawingsthathelpstudentsmakesenseofandkeeptrackoftheunderlyingreasoningandareaneasier

placetostart.Thesevariationsareimportantinitially

forunderstanding.

Butovertime,theselongerwrittenmethodscanbe

abbreviatedintoshorterwrittenmethodsthatarevariationsofwritingthestandardalgorithmforan

operation.

Thislearningpathallowsstudentstoachievefluency

withthestandardalgorithmwhilestillbeingableto

understandandexplaintheshortenedmethod.

Emphasizedmethodsonwhichstudentsspend

significanttimemusthaveaclearlearningpathtosome

writtenvariationofthestandardalgorithm.And

studentswhocanmovetothiswrittenvariationshould

beabletodoso.


10/21

Written Variations of Standard Algorithms

Quantity Model Good Variations Current CommonNew Groups

BelowShow AllTotals

New GroupsAbove

1 1

1 8 9 1 8 9 1 8 9+ 1 5 7 + 1 5 7 + 1 5 7

3 4 6 2 0 0 3 4 61 3 0

1 6

3 4 6

Ungroup Everywhere First,Then Subtract Everywhere

R L Ungroup,Then Subtract,

Ungroup, Then

SubtractLeft Right Right Left

13 13 132 14 16 2 3 16 2 3 163 4 6 3 4 6 3 4 6

- 1 8 9 - 1 8 9 - 1 8 9

1 5 7 1 5 7 1 5 7

Area Model Place ValueSections

Expanded Notation 1-Row

43 = 40 + 3 12

2 4 0 0 67 = 60 + 7 4 3

1 8 0 60 40 = 2 4 0 0 x 6 7

2 8 0 60 3 = 1 8 0 3 0 1+ 2 1 7 40 = 2 8 0 2 5 8

2 8 8 1 7 3 = 2 1 2 8 8 1

2 8 8 1

Rectangle Sections Expanded Notation Digit by Digit

3

4 0 4 3

67 2 8 8 1 67 2 8 8 1

- 2 6 8 0 - 2 6 8

2 0 1 2 0 1

- 2 0 1 - 2 0 1

100

10

x

40 + 3

60+7

2400

280 21

180

40 + 3

67 2 8 8 1- 2 6 8 0 2 0 12 0 1

2 0 1 0

= 43 43

1 1

1

1

) )


11/21

Learning Path for Multidigit Computation in CCSS-MBridgingforteachersandstudents Learning

bycoherentlearningsupports Path

Phase 3: Compact methods forfluencyStudents use any good variation of writing the standard algorithm with

no drawing to build fluency. They explain occasionally to retain meanings.

Phase 2: Research-based mathematically-desirable andaccessible methods in the middle for

understanding and growing fluency

Students focus on and compare efficient, accurate, and generalizablemethods, relating these to visual models and explaining the methods.

They write methods in various ways and discuss the variations.

They may use a helping step method for understanding and/or accuracy.They choose a method for fluency and begin solving with no drawing.

Phase 1: Students methods elicited forunderstanding

but move rapidly to Phase 2Students develop, discuss, and use efficient, accurate, and generalizable

methods and other methods. They use concrete models or drawings thatthey relate to their written method and explain the reasoning used.

Note. Students may consider problems with special structure (e.g., 98 + 76) and devise quick

methods for solving such problems. But the major focus must be on general problems and ongeneralizable methods that focus on single-digit computations (i.e., that are or will generalize to

become a variation of writing the standard algorithm).

Math Drawings

Math Explaining

Math Drawings

Math Explaining

Math Sense-Making

Math Structure

Math Sense-Making

Math Structure


12/21

WhattoEmphasizeandWheretoInterveneasNeeded

GradesKto2aremoreambitiousthansome/manyearlier

statestandards:K:Theteninteennumbers

G1:+within100withcomposinganewten;okifmany

childrenstillusemathdrawings;nosubtraction

withoutdecomposingaten

G2:a)+-total100withcomposinganddecomposinga

ten;usemathdrawingsinitially,butfluency

requiresnomathdrawings

b)+-totals101to1,000withmathdrawings;vitalget

masterybymostsothatG3canfocusonx;

intervenewithasmanyaspossibletogetG2

mastery

Grades3to6arelessambitiousthansome/manyearlierstatestandards:

G3:FluencyforG2goals+-totals101to1,000,sono

mathdrawings;nonewproblemsizessocanfocus

onx[interveneforxallyear]

G4andG5:a)xonlyupto1-digitx2-,3-,4-digitsand2-

digitsx2-digits;sonotreallyneedmasteryof1-rowmethodsformultiplication[havetimeforfractions]

b)divisionhasonlytherelatedunknownfactor

problems;1-digitdivisorsG4and2-digitdivisorsG5;

fluencyG6


13/21

TheComputationLearningPath

Anymethodthatistaughtorusedmusthavealearning

pathrestingonvisualmodelsandonexplainingthereasoningused.Itisnotacceptabletoteachmethods

byrotewithoutunderstandinghowplacevaluesare

usedinthemethods.

Methodsareelicitedfromstudentsanddiscussed,but

goodvariationsofwritingthestandardalgorithmareintroducedearlyonsothatallstudentscanexperience

them.

Stepsinwrittenmethodsareinitiallyrelatedtostepsin

visualmodels.

Experiencinganddiscussingvariationsinwritinga

methodisimportantmathematically.

Studentsstopmakingdrawingswhentheydonotneed

them.Fluencymeanssolvingwithoutadrawing.

StudentsdropstepsofHelpingStepmethodswhenthey

canmovetoashortwrittenvariationofthestandard

algorithmforfluency.


14/21

FIGURE 1: Multidigit Addition Methods that Begin with

One Undecomposed Number (Count or Add On)

A. 456

556

606

616

623

{ {

B. 456 + 167

100 50 10 7

456 556 606 616 623

C. 456 + 100

556 + 50

606 + 10

616 + 7

623

167

67

17

7

0

General Methods for 2 and 3-digit numbers

(shown for 456 + 167)


15/21


16/21


17/21

549

8

8

549 = 500 + 40 + 9

Array/area drawing for 8 549

8 500 =

8 5 hundreds =

40 hundreds

8 40 =

8 4 tens =

32 tens

8 9

= 72

Left to right

showing the

partial products

Right to left

showing the

partial products

Right to left

recording the

carries below

4000

320

72

4392

thinking:

8 5 hundreds

8 4 tens

8 9

549

8

549

8

72

320

4000

4392

thinking:

8 5 hundreds

8 4 tens

8 9 022

4392

34 7

Method A: Method B: Method C:

Method A proceeds from left to right, and the others from right to left.

In Method C, the digits representing new units are written below the line

rather than above 549, thus keeping the digits of the products close to

each other, e.g., the 7 from 89=72 is written diagonally to the left of

the 2 rather than above the 4 in 549.

8 549 = 8 (500 + 40 + 9)= 8 500 + 8 40 + 8 9

Figure 4: Written Methods for the Standard Multiplication Algorithm,

1-digit 3-digit


18/21

Array/area drawing for 36 94

30

+

6

90 + 4

30 90 =

3 tens 9 tens =27 hundreds =

2700

6 90 =

6 9 tens

54 tens =

540

30 4 =

3 tens 4 =12 tens =

120

6 4 = 24

Showing the

partial products

Recording the carries below

for correct place value placement

94

36

94

36

24

540

120

2700

thinking:

3 tens 4

3 tens 9 tens

6 9 tens

6 4

3384

5 2

2 1

1

1

3384

44

720

0 because we

are multiplying

by 3 tens in this row

94

36

3384

2

1

1

564

2820

Method D: Method E:

Method E

variation:

2700

540

120

24

33841

90 + 4

30+

6

2700120

540 24

Area Method F:

Lattice Method G:

2

7

5 4

12

2 4

3

3

3

tens ones

tens

tens

ones

ones

Th

H

1

8

9

4

6

4

Figure 5: Written Methods for the Standard Multiplication Algorithm,

2-digit 2-digit

Written Methods D and E are shown from right to left, but could go from left to right.

In Method E, and Method E variation, digits that represent newly composed tens and hundreds in the partial

products are written below the line intead of above 94. This way, the 1 from 30 4 = 120 is placed correctly in

the hundreds place, unlike in the traditional alternate to Method E, where it is placed in the (incorrect) tens place.

In the Method E variation, the 2 tens from 6 4 = 24 are added to the 4 tens from 6 90 = 540 and then crossed

out so they will not be added again; the situation is similar for the 1 hundred from 30 4 = 120. In Method E, all

multiplying is done rst and then all adding. In the variations following Method E, multiplying and adding alternate,

which is more dicult for some students.

Note that the 0 in the ones place of the second line of Method E is there because the whole line of digits is produced

by multiplying by 30 (not 3). This is also true for the following two methods.

36 94 = (30 + 6)(90 + 4)

= 3090 + 304 + 690 + 64

OR:

94

36

3384

21

1

564

2820

Traditional

alternate to

Method E:


19/21

Figure 6: Further Written Methods for the Standard Multiplication Algorithm,

2-digit 2-digit

Method H: Helping Steps Method I:

A misleading

abbreviated method

94

36

5642821

12

3384

From 30 4 = 120.

The 1 is 1 hundred,

not 1 ten.

94 = 90 + 4

36 = 30 + 6

30 90 = 2700

30 4 = 120

6 90 = 5406 4 = 24

1

3384


20/21

266

- 210

56

7 )966- 700

266

- 210

56

- 56

0

8

30

100

7 )966

138

7 )966- 7

26

- 21

56

- 56

0

138

7

? hundreds + ? tens + ? ones

7

100 + 30 + 8 = 138

966???

966

-700

266

56

-56

0

Area/array drawing for 966 7

Thinking: A rectangle has area 966 and one side of length 7. Find the unknown

side length. Find hundreds rst, then tens, then ones.

Figure 7: Written methods for the standard division algorithm,

1-digit divisor

966 = 7100 + 730 + 78

= 7(100 + 30 + 8)

= 7138

Method A:

Method B:

Conceptual language for this method:

Find the unknown length of the rectangle;

rst nd the hundreds, then the tens, then the ones.

The length gets 1 hundred (units); 2 hundreds (square units) remain.

The length gets 3 tens (units); 5 tens (square units) remain.

2 hundreds + 6 tens = 26 tens.

5 tens + 6 ones = 56 ones.

The length gets 8 ones; 0 remains.

The bringing down steps represent unbundling a remaining

amount and combining it with the amount at the next lower place.


21/21

Figure 8: Written methods for the standard division algorithm,

2-digit divisor

305

- 270

35

27 )1655-1350

305

- 270

35-27

8

1

10

5061

61

27 )1655- 162

35

- 27

8

5127 )1655

- 135

30

- 27

35

- 27

8

27

50 + 10 + 1 = 61

1655

- 1350

305

35

- 27

8

Method A:

Method B: Two variations

165527

(30)

(30) Rounding 27 toproduces the

underestimate

50 at the rst

step, but this

method allows

the division

process to becontinued.

Erase an underestimate to

make it exact.

Change the multiplier but not the

underestimated product; then

subtract more.

6

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st alg? assm + ncsm talks 2013

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